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Doob h-Transform

A Doob h-transform is a change of measure that constructs a new Markov process by reweighting transitions of an original process using a positive eigenfunction h. If (P_t)_(t>=0) is the transition semigroup of the original process and its infinitesimal generator L satisfies Lh=lambdah, the transformed semigroup is

P_t^hf(x)=e^(-lambdat)(P_t(hf)(x))/(h(x)).
(1)

Its infinitesimal generator is therefore

L^hf=(L(hf))/h-lambdaf.
(2)

When the process has transition densities p_t(x,y), the transformed densities are

p_t^h(x,y)=e^(-lambdat)(h(y))/(h(x))p_t(x,y).
(3)

The case lambda=0 uses a positive harmonic function h. Space-time versions of the construction condition a process on a future endpoint. Taking h_t(x)=p_(T-t)(x,b) produces the corresponding Markov bridge to b at time T.

Doob (1957) used this transformation to construct conditioned Brownian motion and relate it to boundary limits of harmonic functions.

See also

Brownian Bridge, Conditional Probability, Markov Bridge

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References

Doob, J. L. "Conditional Brownian Motion and the Boundary Limits of Harmonic Functions." Bull. Soc. Math. France 85, 431-458, 1957. https://doi.org/10.24033/bsmf.1494.Fitzsimmons, P. J.; Pitman, J.; and Yor, M. "Markovian Bridges: Construction, Palm Interpretation, and Splicing." In Seminar on Stochastic Processes, 1992. Boston, MA: Birkhäuser, pp. 101-134, 1993. https://doi.org/10.1007/978-1-4612-0339-1_5.

Cite this as:

Weisstein, Eric W. "Doob h-Transform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Doobh-Transform.html

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